Perl 5, 11692 bytes
sub$r=$_;
$w=$r*2+11;
$_="P1 f{$r=pop;$w=$r*2+11;$s=0$w x$w**2;substr$s,sin$w @{[map{($x$_%$w-$r-5)*$r+$r+5+int(cos**2+($x)*$r+$r+5$_/$w-$r-5)*$w,**2<$r**2?1,1while++$x<1e5;"P1 $w $w $s":0}0..$w**2-1]}"
Just a shorter translation of the C answer from @ErikFCircle with black filling. Except not caring about keeping x below 2πPut the 92 bytes above into program.pl and run like this:
echo 50 | perl -p program.pl > circle.pbm # radius 50
feh circle.pbm # view with feh or other image viewer
Perl 5, 116 bytes
sub f{$r=pop;$w=$r*2+11;$s=0 x$w**2;substr$s,sin($x)*$r+$r+5+int(cos($x)*$r+$r+5)*$w,1,1while++$x<1e5;"P1 $w $w $s"}
Just a shorter translation of the C answer from @ErikF. Except not caring about keeping x below 2π.
Perl 5, 92 bytes
$r=$_;
$w=$r*2+11;
$_="P1 $w $w @{[map{($_%$w-$r-5)**2+($_/$w-$r-5)**2<$r**2?1:0}0..$w**2-1]}"
Circle with black filling. Put the 92 bytes above into program.pl and run like this:
echo 50 | perl -p program.pl > circle.pbm # radius 50
feh circle.pbm # view with feh or other image viewer